On some Diophantine equations involving balancing numbers

summary:In this paper, we find all the solutions of the Diophantine equation $B_1^p+2B_2^p+\cdots +kB_k^p=B_n^q$ in positive integer variables $(k, n)$, where $B_i$ is the $i^{th}$ balancing number if the exponents $p$, $q$ are included in the set $\lbrace 1,2\rbrace$

On repdigits as product of consecutive Fibonacci numbers

Let (F$_{n}$)$_{n\geq0}$ be the Fibonacci sequence. In 2000, F. Luca proved that F10 = 55 is the largest repdigit (i.e. a number with only one distinct digit in its decimal expansion) in the Fibonacci sequence. In this note, we show that if Fn · · · F$_{n+(k-1)}$ is a repdigit, with at least two digits, then (k, n) = (1, 10)

On the Ramanujan-Nagell type Diophantine equation (Dx^2+k^n=B)

In this paper, we prove that the Ramanujan-Nagell type Diophantine equation (Dx^2+k^n=B) has at most three nonnegative integer solutions ((x, n)) for (k) a prime and (B, D) positive integers

On two Diophantine equations of Ramanujan-Nagell type

In this paper, we prove two conjectures of Ulas ([21]) on two Diophantine equations of Ramanujan-Nagell type. In fact, we show that the following equations x2+(2m+1+1)2n=24(m+1)+23(m+1)+22m+2m+1+1, x2+(22m+6-1)2n/3 = (49 · 42m+5-11· 4m+3+1)/9 have exactly four solutions

Lucas factoriangular numbers

summary:We show that the only Lucas numbers which are factoriangular are $1$ and $2$

On a family of two-parametric D(4)-triples

Let k be a positive integer. In this paper, we study a parametric family of the sets of integers {k,A2k+4A,(A+1)2k+4(A+1),d}. We prove that if d is a positive integer such that the product of any two distinct elements of that set increased by 4 is a perfect square, then d= (A4 + 2A3 + A2)k3 + (8A3 + 12A2 + 4A)k2 + (20A2 + 20A + 4) k + (16A + 8) for 1≤ A ≤22 and A ≥ 51767